Assuming that the process \(\mathrm{ve} \mathrm{e}^{-} \rightarrow \mathrm{e}^{-} \mathrm{v}_{\mathrm{e}}\) occurs only by the weak charged- current interaction (i.e. ignoring the Z-exchange neutral-current process), show that $$ \sigma_{c c}^{v_{e} e^{-}} \approx \frac{2 m_{\mathrm{e}} E_{v} \sigma_{f}^{2}}{\pi} $$ where \(E_{v}\) is neutrino energy in the laboratory frame in which the struck \(e^{-}\)is at rest.

Neutrino-electron scattering is an important phenomenon in particle physics.
It involves the interaction between a neutrino and an electron, usually mediated by the weak force.
In our specific problem, we are considering the scattering of an electron neutrino \(u_e\) with an electron \(e^{-}\).
This interaction is crucial for understanding processes in the early universe and in particle detectors.
The interaction is described by the process \(u_e e^{-} \rightarrow e^{-} u_e\).
In this case, we are focusing on the **weak charged-current interaction** only.
This kind of interaction involves the exchange of a W boson.
Ignoring other possible interactions, like the neutral-current process mediated by a Z boson, simplifies the analysis.
This ensures our equations and calculations are more straightforward.

Fermi’s Golden Rule is a principle used to calculate the transition rate (or probability) of particles changing states.
Here, it helps us determine the cross-section for our neutrino-electron scattering.
Fermi's Golden Rule states that the cross-section \(\sigma\) is proportional to:

• The square of the matrix element \( |\mathcal{M}|^2 \)
• A phase space factor that accounts for the final states of the particles

In simpler terms, the more likely the states are able to match in the final condition, the higher the cross-section will be.
For our interaction, we use this to outline how likely it is for the neutrino to scatter off the electron.

The matrix element \( \mathcal{M} \) represents the probability amplitude of the interaction.
For weak interactions like ours, the W boson mediates the force.
The matrix element for this specific weak charged-current interaction is proportional to Fermi's coupling constant \(G_F \).
This simplifies the equations since \(G_F \) is a well-known constant.
The matrix element squared \( |\mathcal{M}|^2 \) then becomes:

• Proportional to \( G_F^2 \)
• Multiplied by the neutrino energy \( E_{u} \)
• Multiplied by the electron mass \( m_e \)

Thus, we can write:
\[ |\mathcal{M}|^2 \propto G_F^2 E_{u} m_e \]
Now, we have a key component needed for our cross-section calculation.

Combining our previous steps, we can determine the cross-section \( \sigma \) for the neutrino-electron interaction.
The cross-section tells us how likely this scattering event is to occur.
From Fermi's Golden Rule, we know that \( \sigma \) is proportional to the squared matrix element and a phase space factor.
The phase space factor, when calculated, contributes a term proportional to the neutrino energy \(E_{u}\).
Thus, the cross-section \( \sigma_{cc}^{u_{e} e^{-}} \) becomes:
\[ \sigma_{cc}^{u_{e} e^{-}} \approx \frac{2 m_e E_{u} G_F^2}{\pi} \]
Finally, substituting \( G_F^2 = \sigma_{f}^{2} \) converts our equation to:
\[ \sigma_{cc}^{u_{e} e^{-}} \approx \frac{2 m_{e} E_{u} \sigma_{f}^{2}}{\pi} \]
This complete expression shows the dependency on electron mass, neutrino energy, and Fermi's coupling constant.